Snapping mesh together by a point without stretching the object
On the object you want to move move the origin to the face that should line up with the other object. In Edit Mode select the face and use Snap Cursor to Selected That will move the 3D cursor to the center of the face. Then switch to object mode and do Transform Origin to 3D Cursor.



Next, select the other object and, in Edit Mode, snap the cursor to the face you want to snap to. Exit edit mode, select the previous object (the hemisphere) and do Snap Selection to Cursor.

(The shortcut for the Snap tool is ShiftS